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Yang–Mills for Mathematicians Infosys-ICTS Ramanujan Lectures: Lecture 1 Yang{Mills for mathematicians Infosys-ICTS Ramanujan Lectures: Lecture 1 Sourav Chatterjee Sourav Chatterjee Yang{Mills for mathematicians What is this talk about? I I am going to give you a heavily compressed exposition of a very long story, and four open problems. I There will be reading references at the end, if you want to seriously learn about it. I Physicists are generally familiar with most of what I'm going to say, but mathematicians are not. This talk is for mathematicians. Sourav Chatterjee Yang{Mills for mathematicians Quantum field theories I Quantum field theories explain interactions between elementary particles and make predictions about their behaviors. I Encapsulated by the Standard Model. I Yang{Mills theories are certain kinds of important quantum field theories that constitute the standard model. I What is a QFT? | This is an open question, not only in mathematics, but also in physics. I Remarkably, physicists can calculate and make surprisingly accurate predictions using QFTs, without really understanding what these objects are! I The mathematical construction of quantum field theories | more specifically Yang{Mills theories | is one of the seven millennium problems posed by the Clay Institute. Sourav Chatterjee Yang{Mills for mathematicians QFT basics 4 I Spacetime: R . Restricted Lorentz transforms: A group of 4 linear transformations of R . I Poincar´egroup P consists of all (a; Λ), where Λ is a restricted 4 Lorentz transformation and a 2 R . 4 I P acts on R as (a; Λ)x = a + Λx. I Special relativity: The laws of physics remain invariant under change of coordinates by the action of the Poincar´egroup. I A quantum field theory models the behavior of a physical system (e.g. a collection of elementary particles) using: I a Hilbert space H, and I a (projective) unitary representation U of P in H. I Assumptions: I To an observer, the state of the system appears as some vector 2 H. If is known, we can compute probabilities of events. I To a different observer, who is using a coordinate system obtained by the action of (a; Λ) on the coordinate system of the first observer, the state appears as U(a; Λ) . Sourav Chatterjee Yang{Mills for mathematicians Time evolution I Suppose a stationary observer at spatial location (0; 0; 0) observes the physical system in state at time 0. I After time has evolved by t, the observer is in a new coordinate system obtained by the action of ((−t; 0; 0; 0); Id.) on the original coordinate system. (The spacetime point (t; x; y; z) in the original coordinate system becomes (0; x; y; z) in the new coordinate system.) I Thus, after time t, the system will appear to the observer as being in state U((−t; 0; 0; 0); Id.) . I It can be proved that there is a self-adjoint operator H on H so that for any t, U((−t; 0; 0; 0); Id.) = e−itH . I H is called the Hamiltonian. I If a different observer was at (x; y; z) at time 0, and moves with constant velocity v for time t, he will observe the system to be in state U((−t; x; y; z); Λ) , where Λ is a restricted Lorentz transformation depending on v. Sourav Chatterjee Yang{Mills for mathematicians Quantum field I Suppose we are given H and U. 4 I A quantum field ' is a hypothetical function on R , which 4 when integrated against a smooth test function on R , yields an operator on H. I To put it more succinctly, it is an operator-valued distribution. I The quantum field ' related to our physical system is a field that satisfies '(a + Λx) = U(a; Λ)'(x)U(a; Λ)−1: I The field ' is used for calculating probabilities of events and expected values of various observables. In fact, it becomes the central object of interest in the study of the system. Sourav Chatterjee Yang{Mills for mathematicians Wightman axioms I The most popular approach to giving a fully rigorous definition of a quantum field theory is via the Wightman axioms. I These axioms are essentially a more precise version of what I described in the previous slides. I They include some additional conditions (such as `locality') that must be satisfied by H, U and ', and some assumptions about the existence and properties of a unique vacuum state Ω 2 H of our system. (This the lowest eigenstate of H.) I The axioms give the bare minimum conditions required to avoid physical inconsistencies. I It has been possible to construct certain simple QFTs, known as free fields, which satisfy the Wightman axioms. I Free fields describe trivial systems of particles that do not interact with each other. I Unfortunately, no one has been able to rigorously construct a nontrivial (interacting) QFT satisfying the Wightman axioms. Sourav Chatterjee Yang{Mills for mathematicians Calculations??? I If we cannot even define the theory, how can we calculate? I Physicists get around this problem by doing perturbative expansions around free fields. I That is, they assume that the desired QFT is a `small perturbation' of the free field (which is well-defined), and do a kind of Taylor expansion around it. I The calculations involve Feynman diagrams and renormalization. I However, there is a rigorous theorem due to Haag, which says that the Hilbert space for an interacting theory cannot be the same as the Hilbert space for a non-interacting theory. I So it is not clear how one can justify such a perturbative expansion. In fact, in most cases it is not clear what the new Hilbert space is! I And yet, in many cases, these calculations yield results that match experiments to remarkable degrees of accuracy. Sourav Chatterjee Yang{Mills for mathematicians The probabilistic approach (Constructive QFT) I There is a probabilistic approach to constructing QFTs that satisfy the Wightman axioms. It goes as follows: 4 I First, construct a random field ξ on R whose probability law is related to the desired QFT in a certain way. Usually this is a random distribution, and not a random function. I ξ is called a Euclidean QFT. I Show that ξ satisfies a set of conditions known as the Osterwalder{Schrader axioms. I If this is true, then there is a reconstruction theorem that allows us to construct the desired QFT (i.e., H, U, ' and Ω.) I In general, the QFT is nontrivial if and only if the field ξ is non-Gaussian. I The program, initiated in the 60s, was successful in constructing nontrivial QFTs when the dimension of spacetime was reduced from 4 to 2 or 3 | but not yet in dimension 4. 4 I The most notable achievements were the constructions of '2 4 and '3 theories (in spacetime dimensions 2 and 3, respectively). Sourav Chatterjee Yang{Mills for mathematicians Yang{Mills theories 4 I ' theories are mathematically interesting, but describe no real physical system. I To venture into the real world, one has to consider 4D Yang{Mills theories. I These are QFTs that describe interactions between real elementary particles. I The question is completely settled in 2D. I There was a tremendous amount of work on rigorously constructing Yang{Mills theories in 3D and 4D, by Ba laban and others. I However, the investigation was inconclusive and the question is still considered to be open. I Even the first step in the probabilistic approach, namely, the construction of a random field, remains open. We will now talk about that. Sourav Chatterjee Yang{Mills for mathematicians Euclidean Yang{Mills theories I Recall that the first step in the probabilistic approach to constructing QFTs is the construction of a suitable random field, known as a Euclidean QFT. I For Yang{Mills theories, these random fields are called Euclidean Yang{Mills theories. I These have not yet been constructed in spacetime dimensions 3 and 4. I Euclidean Yang{Mills theories are supposed to be scaling limits of lattice gauge theories, which are well-defined discrete probabilistic objects, which I will now discuss. Sourav Chatterjee Yang{Mills for mathematicians Lattice gauge theories I Let d = dimension of spacetime, and G be a matrix Lie group. (Most important: d = 4 and G = SU(2) or SU(3).) I The lattice gauge theory with gauge group G on a finite set d Λ ⊆ Z is defined as follows. I Suppose that for any two adjacent vertices x; y 2 Λ, we have a group element U(x; y) 2 G, with U(y; x) = U(x; y)−1. I Let G(Λ) denote the set of all such configurations. I A square bounded by four edges is called a plaquette. Let P(Λ) denote the set of all plaquettes in Λ. I For a plaquette p 2 P(Λ) with vertices x1; x2; x3; x4 in anti-clockwise order, and a configuration U 2 G(Λ), define Up := U(x1; x2)U(x2; x3)U(x3; x4)U(x4; x1): I The Wilson action of U is defined as X SW(U) := Re(Tr(I − Up)): p2P(Λ) Sourav Chatterjee Yang{Mills for mathematicians Definition of lattice gauge theory contd. I Let σΛ be the product Haar measure on G(Λ). I Given β > 0, let µΛ,β be the probability measure on G(Λ) defined as 1 dµ (U) := e−βSW(U)dσ (U); Λ,β Z Λ where Z is the normalizing constant. I This probability measure is called the lattice gauge theory on Λ for the gauge group G, with inverse coupling strength β. I An infinite volume limit of the theory is a weak limit of the d above probability measures as Λ " Z (may not be unique).
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